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unknown, or in some sense transcendental as compared with 

 + , so that it would be impossible to express a term affected 

 with the sign + ^ by means of any combination of terms affected 

 with the sign -f-. With this understanding we might call 

 terms affected with +* imaginary, and the importance of the 

 distinction between reals and imaginaries consists in this: that 

 when we have an equation, u = 0, involving quantities of 

 both kinds, we may put the real and imaginary parts of u 

 respectively equal to 0. 



The following example will show the advantage of employ- 

 ing such a symbol. If we put +^0 in place of + ^ in the 

 development of e*, we find 



e *■ = to^^ +* stn^ (1) 



where cosi^ and ^m^ respectively stand for the series 



^ + 1.2+ 1.2.3.4 + ^''' 



^ +17273+ 1 .2.3.4.5 +^^- 

 multiplying equation (1) by the similar one 



we have 



But again, from (1) we have 



And since we are entitled to compare the real and imaginary 

 parts in the two last equations, we conclude that 



I 



